This applet looks at combining two risky investments, S and B, [br]in a risky portfolio, P, where each investment, S and B, has an expected rate of return, [br]E(r), and a level of risk, [math]\sigma[/math]. [br]Risky asset S can be thought of as a portfolio of all stock in the economy and riskyasset B can be thought of as a portfolio of all risky bonds in the economy. [br]Besides the expected return and risk of the investments, we must also know the[br]correlation, [math]\rho_{S,B}[/math] (or covariance, [math]\sigma_{S,B}[/math]) between the returns of the two risky assets [br]and the investment weighting each risky asset is given in the portfolio, P. [br]We then compute the rate of return and the level of risk for P, the risky portfolio. [br]Changing the weight in risky asset S, w[sub]s[/sub], moves P along the risk-return curve. [br]Changing the features (i.e. expected returns, levels of risk, [u]and[/u] correlation[br]coefficient of the returns for the two risky assets) of S and B moves the risk-return[br]curve itself. Pay special attention to the impact changing the correlation[br]coefficient has on the shape of the risk-return curve.[br][br]

The basic assumption is that portfolio P is achieved by investing in each risky asset, [br]S and B, such that the weights of investment sum to 1.  Then[br]the expected return for portfolio P is a weighted average of the rates of[br]return on S and B (i.e. E(r[sub]p[/sub]) = w[sub]S[/sub]*E(r[sub]S[/sub])+(1-w[sub]S[/sub])*E(r[sub]B[/sub])).[br][br]If [math]\sigma_S[/math]S is the risk associated with S, and [math]\sigma_B[/math] is the risk associated with B, [br]and r is the correlation coefficient of the returns on the two risky assets, then:[br][math]\sigma^2_P=\left(w_S\cdot\sigma_S\right)^2+\left(w_B\cdot\sigma_B\right)^2+2\cdot\left(w_S\cdot\sigma_S\right)\cdot\left(w_B\cdot\sigma_B\right)\cdot\rho_{S,B}[/math][br] Note: the above equation gives then variance of returns on the risky portfolio, P. [br]To get the standard deviation, [math]\sigma_P[/math], we simply take the square root of variance.[br]Notice that changing w moves portfolioP along the risk-return curve, while changing the parameters of the two risky[br]assets (including the correlation coefficient) moves the curve itself.[br]

Consider what happens if E(r[sub]S[/sub])= 0.20, [math]\sigma_S[/math] = 0.30, E(r[sub]B[/sub]) = 0.10, and [math]\sigma_B[/math]= 0.15. [br]Now consider weights of 50% in each asset. Notice that a simple weighted average for the[br]riskiness of portfolio P would be [math]\sigma_P[/math]= 0.5* 0.30 + 0.50*0.15 = 0.225. As long as [math]\rho_{S,P}[/math] is less than 1, [br]the riskiness of P is less than 0.225. This is diversification – the elimination of risk! [br][br]When  [math]\rho_{S,P}[/math]=1 this suggests the returns of S and B are perfectly correlated. [br]The resulting risk-return curve is a simple line segment, suggesting zero benefit from diversification. [br][br]If [math]\rho_{S,P}[/math] = -1 this suggests there is a perfect correlation between the upside of one investment [br]and the downside of the other. In other words, the risky assets are perfectly negatively[br]correlated. In this scenario, there exists a zero-risk portfolio with an[br]abnormal return (i.e. the portfolio that intersects the y-axis). [br]